File "Barrett.php"
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<?php
/**
* BCMath Barrett Modular Exponentiation Engine
*
* PHP version 5 and 7
*
* @author Jim Wigginton <terrafrost@php.net>
* @copyright 2017 Jim Wigginton
* @license http://www.opensource.org/licenses/mit-license.html MIT License
* @link http://pear.php.net/package/Math_BigInteger
*/
namespace phpseclib3\Math\BigInteger\Engines\BCMath\Reductions;
use phpseclib3\Math\BigInteger\Engines\BCMath\Base;
/**
* PHP Barrett Modular Exponentiation Engine
*
* @author Jim Wigginton <terrafrost@php.net>
*/
abstract class Barrett extends Base
{
/**
* Cache constants
*
* $cache[self::VARIABLE] tells us whether or not the cached data is still valid.
*
*/
const VARIABLE = 0;
/**
* $cache[self::DATA] contains the cached data.
*
*/
const DATA = 1;
/**
* Barrett Modular Reduction
*
* See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} /
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information. Modified slightly,
* so as not to require negative numbers (initially, this script didn't support negative numbers).
*
* Employs "folding", as described at
* {@link http://www.cosic.esat.kuleuven.be/publications/thesis-149.pdf#page=66 thesis-149.pdf#page=66}. To quote from
* it, "the idea [behind folding] is to find a value x' such that x (mod m) = x' (mod m), with x' being smaller than x."
*
* Unfortunately, the "Barrett Reduction with Folding" algorithm described in thesis-149.pdf is not, as written, all that
* usable on account of (1) its not using reasonable radix points as discussed in
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=162 MPM 6.2.2} and (2) the fact that, even with reasonable
* radix points, it only works when there are an even number of digits in the denominator. The reason for (2) is that
* (x >> 1) + (x >> 1) != x / 2 + x / 2. If x is even, they're the same, but if x is odd, they're not. See the in-line
* comments for details.
*
* @param string $n
* @param string $m
* @return string
*/
protected static function reduce($n, $m)
{
static $cache = [
self::VARIABLE => [],
self::DATA => []
];
$m_length = strlen($m);
if (strlen($n) > 2 * $m_length) {
return bcmod($n, $m);
}
// if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reduced
if ($m_length < 5) {
return self::regularBarrett($n, $m);
}
// n = 2 * m.length
if (($key = array_search($m, $cache[self::VARIABLE])) === false) {
$key = count($cache[self::VARIABLE]);
$cache[self::VARIABLE][] = $m;
$lhs = '1' . str_repeat('0', $m_length + ($m_length >> 1));
$u = bcdiv($lhs, $m, 0);
$m1 = bcsub($lhs, bcmul($u, $m));
$cache[self::DATA][] = [
'u' => $u, // m.length >> 1 (technically (m.length >> 1) + 1)
'm1' => $m1 // m.length
];
} else {
extract($cache[self::DATA][$key]);
}
$cutoff = $m_length + ($m_length >> 1);
$lsd = substr($n, -$cutoff);
$msd = substr($n, 0, -$cutoff);
$temp = bcmul($msd, $m1); // m.length + (m.length >> 1)
$n = bcadd($lsd, $temp); // m.length + (m.length >> 1) + 1 (so basically we're adding two same length numbers)
//if ($m_length & 1) {
// return self::regularBarrett($n, $m);
//}
// (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2
$temp = substr($n, 0, -$m_length + 1);
// if even: ((m.length >> 1) + 2) + (m.length >> 1) == m.length + 2
// if odd: ((m.length >> 1) + 2) + (m.length >> 1) == (m.length - 1) + 2 == m.length + 1
$temp = bcmul($temp, $u);
// if even: (m.length + 2) - ((m.length >> 1) + 1) = m.length - (m.length >> 1) + 1
// if odd: (m.length + 1) - ((m.length >> 1) + 1) = m.length - (m.length >> 1)
$temp = substr($temp, 0, -($m_length >> 1) - 1);
// if even: (m.length - (m.length >> 1) + 1) + m.length = 2 * m.length - (m.length >> 1) + 1
// if odd: (m.length - (m.length >> 1)) + m.length = 2 * m.length - (m.length >> 1)
$temp = bcmul($temp, $m);
// at this point, if m had an odd number of digits, we'd be subtracting a 2 * m.length - (m.length >> 1) digit
// number from a m.length + (m.length >> 1) + 1 digit number. ie. there'd be an extra digit and the while loop
// following this comment would loop a lot (hence our calling _regularBarrett() in that situation).
$result = bcsub($n, $temp);
//if (bccomp($result, '0') < 0) {
if ($result[0] == '-') {
$temp = '1' . str_repeat('0', $m_length + 1);
$result = bcadd($result, $temp);
}
while (bccomp($result, $m) >= 0) {
$result = bcsub($result, $m);
}
return $result;
}
/**
* (Regular) Barrett Modular Reduction
*
* For numbers with more than four digits BigInteger::_barrett() is faster. The difference between that and this
* is that this function does not fold the denominator into a smaller form.
*
* @param string $x
* @param string $n
* @return string
*/
private static function regularBarrett($x, $n)
{
static $cache = [
self::VARIABLE => [],
self::DATA => []
];
$n_length = strlen($n);
if (strlen($x) > 2 * $n_length) {
return bcmod($x, $n);
}
if (($key = array_search($n, $cache[self::VARIABLE])) === false) {
$key = count($cache[self::VARIABLE]);
$cache[self::VARIABLE][] = $n;
$lhs = '1' . str_repeat('0', 2 * $n_length);
$cache[self::DATA][] = bcdiv($lhs, $n, 0);
}
$temp = substr($x, 0, -$n_length + 1);
$temp = bcmul($temp, $cache[self::DATA][$key]);
$temp = substr($temp, 0, -$n_length - 1);
$r1 = substr($x, -$n_length - 1);
$r2 = substr(bcmul($temp, $n), -$n_length - 1);
$result = bcsub($r1, $r2);
//if (bccomp($result, '0') < 0) {
if ($result[0] == '-') {
$q = '1' . str_repeat('0', $n_length + 1);
$result = bcadd($result, $q);
}
while (bccomp($result, $n) >= 0) {
$result = bcsub($result, $n);
}
return $result;
}
}